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Here we are assuming that the badness of rain, if you leave the window off, would be twice the goodness of going to the party, and that not going to the party would be neither bad nor good (it does not matter that we used -10 and +5 to represent this relationship; we could just as well have used -4 and +2, or -2 and +1). In this situation, then, you should stay home, unless the probability of its raining is less than half the probability of its not raining. That is, you should stay home unless the probability of rain is less than one third.

What we are leading up to is the concept of expected value:

Let o1; o2... be the possible outcomes of an action A; let V(o) be the value (cost or benefit) of each outcome o, and let P(o) be the probability of each outcome (given that action A was performed). Then the expected value of an action A is:

That is, for each possible outcome of the action, you multiply the probability of the outcome by its value (its cost or benefit, as the case may be). Then you add these figures together to get the expected value of the action. The idea behind this is that, given a range of possible actions, one should do whatever maximises expected value. If one's possible actions are Av A„. . ., and one of these - say Ak - has the highest expected value, then one should perform Ak. In the case of the window and the party, the possibilities are: stay and repair the window, or go to the party. If we assume that the probability of rain is 0.5, then the expected value of going to the party, according to the values given in the chart above, would be

The expected value of not going to the party is

Since the expected value of staying to repair the window (0) is greater than that of going to the party (-2.5), you should stay to repair the window.

It is very helpful to have a firm grasp of the concept of expected value, because it is one of the areas in which people most frequently make mistakes in reasoning. For example, we sometimes see arguments like this:

The bottom line is this: no matter how safe the government says it is, they cannot rule out the possibility of a catastrophic accident. We should decommission all nuclear power plants as soon as possible.

If the arguer is right, then the expected value of decommissioning all nuclear plants should be greater than that of not doing so. But, although this may be true, the arguer has given us no reason to believe it. In order to reach that conclusion, we would need to know the actual probability of an accident; we would also need to know how the cost of an accident would compare with the benefits and other costs of continuing to use nuclear power, as well as the benefits and costs of relying on other sources of energy. The arguer has provided none of this, and the mere fact that an accident is not impossible is not disputed by anyone. Nor does anyone dispute that a nuclear accident would be a very bad thing. But the mere fact that something bad - no matter how bad - is a possible outcome of some action certainly does not establish that the action should not be performed.

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